Grunsky Inequalities Research Articles

Analysis on quasidisks: A unified approach through transmission and jump problems

We give an exposition of results from a crossroad between geometric function theory, harmonic analysis, boundary value problems, and approximation theory, which characterize quasicircles. We will specifically expose the interplay between the jump decomposition, singular integral operators, approximation by Faber series, and the Grunsky inequalities. Our unified point of view is made possible by the concept of transmission.

A Shimorin-Type Estimate for Higher-Order Schwarzian Derivatives

Using Grunsky inequalities, an estimate on the norm of the multiplication by higher-order Schwarzian derivatives in weighted Bergman spaces is given.

On analytic families of conformal maps

Let Λ be a domain in C and let fλ(z) = z + a0(λ) + a1(λ)z −1 + ... be meromorphic in D∗ := {z ∈ C : |z| > 1} ∪ {∞}. We assume that fλ(z) is holomorphic in λ ∈ Λ for fixed z. The main theorem states: Let Λ0 be a subdomain of Λ such that fλ is univalent in D∗ for λ ∈ Λ0. If fλ0 has a quasiconformal extension to the closure of D∗ for one λ0 ∈ Λ0 then fλ has a quasiconformal extension for all λ ∈ Λ0. This result is related to a theorem of Mane, Sad and Sullivan (1983) where the assumptions are however different. The main tool of our proof is the Grunsky inequality for univalent functions.

Faber–Tietz Functions and Grunsky Coefficients for Mappings into a Torus

Tietz defined a generalization of Faber polynomials for conformal maps into a compact Riemann surface, which we will call the Faber–Tietz functions. We give a generating function for the Faber–Tietz functions of any torus in terms of the Weierstrass zeta function. We also give explicit expressions for the first few Faber–Tietz functions on the torus in terms of the Weierstrass $$\wp $$ -function and Eisenstein series. We also prove a sharp generalization of the Grunsky inequalities for conformal maps into compact Riemann surfaces. We also derive explicit expressions for some of the Grunsky coefficients of a conformal map into a torus.

On Grunsky inequalities and the exact domain of variability of the Grunsky functional

We give a generalized version of the classical Grunsky inequalities. As an application, we study the exact domain of variability of the Grunsky functional for univalent functions with quasiconformal extensions.

Grunsky inequalities and quasiconformal extension

The Grunsky coefficient inequalities play a crucial role in various problems and are intrinsically connected with the integrable holomorphic quadratic differentials having only zeros of even order. For the functions with quasi-conformal extensions, the Grunsky constant ℵ(f) and the extremal dilatationk(f) are related by ℵ(f)≤k(f). In 1985, Jurgen Moser conjectured that any univalent functionf(z)=z+b 0+b 1 z −1+… on Δ*={|z|>1} can be approximated locally uniformly by functions with ℵ(f)<k(f). In this paper, we prove a theorem confirming Moser’s conjecture, which sheds new light on the features of Grunsky coefficients.

Inversion of Ahlfors and Grunsky Inequalities

Abstract We solve the old Kühnau's problem on the exact lower bound in the inverse inequality estimating the dilatation of a univalent function by its Grunsky norm and in the related Ahlfors inequality for Fredholm eigenvalues.

Beyond Moser's Conjecture on Grunsky Inequalities

We prove a weakened version of a conjecture concerning the relation between the Grunsky constant and the Teichmller norm of extensions of univalent functions.

Chordal Loewner families and univalent Cauchy transforms

We study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transform if and only if it belongs to some chordal Loewner family. If the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the Cauchy transform, the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal Cauchy transform, and the second in terms of moment inequalities corresponding to the Grunsky inequalities.

Polygonal quasiconformal maps and grunsky inequalities

There is a deep connection between the Grunsky coefficient inequalities for univalent functions and related extremal quasiconformal maps. In this paper, we develop the technique based on the Grunsky inequalities and apply it to solving a problem concerning polygonal quasiconformal maps.

Conformal invariants and higher-order schwarz lemmas

We derive a generalization of the Grunsky inequalities using the Dirichlet principle. As a corollary, sharp distortion theorems for bounded univalent functions are proven for invariant differential expressions which are higher-order versions of the Schwarzian derivative. These distortion theorems can be written entirely in terms of conformai invariants depending on the derivatives of the hyperbolic metric, and can be interpreted as ’Schwarz lemmas’. In particular, sharp estimates on distortion of the derivatives of geodesic curvature of a curve under bounded univalent maps are given.

On the Teichmüller-Kühnau Extension of Univalent Functions

Abstract We give a complete description of the class of univalent holomorphic functions with regular quasiconformal extensions for which the Grunsky inequalities are both necessary and sufficient. This concerns the question posed by several authors.

Orthogonal decomposition related to magnetic field, and Grunsky inequality

Orthogonal decomposition related to magnetic field, and Grunsky inequality

A variational method for univalent functions connected with antigraphy

The paper is devoted to a class of functions analytic and univalent in the unit disk that are connected with an antigraphy $e^{iφ}\bar{ω} + iρe^{iφ/2}$. Variational formulas and Grunsky inequalities are derived. As an application there are given some esti

Estimates for coefficients of univalent functions from integral means and Grunsky inequalities

Sharp bounds are obtained for the coefficients of inverses of univalent functions in the class Σ(p) by using results on integral means and generalized Grunsky inequalities. A new and elementary proof is given for a result due to Lowner about sharp bounds for coefficients of inverses of functions in the classS.

Grunsky's inequality and its applications to Teichmüller spaces

Grunsky's inequality and its applications to Teichmüller spaces

On the Schwarzian coefficients of univalent functions

For f ∈ S, we study the Schwarzian coefficients sn defined by {f, z} = Σ snzn. Sharp bounds on s0, s1 and s2 are given, together with an order of growth estimate as n → ∞. We use the Grunsky Inequalities to estimate combinations of coefficients.

Sharpened forms of the Grunsky inequalities

Sharpened forms of the Grunsky inequalities

On a class of mappings being quasiconformal in the mean

A generalized area theorem for quasiconformal mappings is proved. Furthermore. modified Grunsky's inequalities are derived for a class (c σ) of mappings being quasiconformal in the mean.

Generalized Grunsky coefficients and inequalities

The generalized Grunsky coefficients are defined in this paper for all locally univalent meromorphic functions in any domain in the complete complex plane. Various explicit formulas for these coefficients are established. Necessary conditions for univalence are obtained in arbitrary domains and in the unit disc in particular. The first one generalizes Grunsky inequalities and the second one is an extension of the Nehari-Schwarzian derivative condition.